She felt like crying, but she couldn’t. She was NOT going to succumb to her tears.

On the other hand, another reason this girl could not cry would be the she was so overwhelmed by the situation she was completely numb, in shock. In this case, her feelings of wanting to cry are so overpowering she finds her normal emotional responses have stopped working. In this latter case, you might write something like this:

She felt desperately like crying, but she couldn’t. Her tears had just seemed to dry up.

… Her tears just weren’t there.

… Her tears, her head, her heart had shut down.

Does this help?

Use a thesaurus if you want to use more unique words.

Succumb means yield or surrender. In your sentence this would mean that the tears would not stop. This is the opposite of what you seem to be trying to say.

Come, start, fall, appear, materialize are all appropriate alternatives.

You can eliminate “but she couldn’t”